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Theorem mapssfset 8613
Description: The value of the set exponentiation (𝐵m 𝐴) is a subset of the class of functions from 𝐴 to 𝐵. (Contributed by AV, 10-Aug-2024.)
Assertion
Ref Expression
mapssfset (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵}
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem mapssfset
StepHypRef Expression
1 mapfset 8612 . . 3 (𝐵 ∈ V → {𝑓𝑓:𝐴𝐵} = (𝐵m 𝐴))
2 eqimss2 3982 . . 3 ({𝑓𝑓:𝐴𝐵} = (𝐵m 𝐴) → (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵})
31, 2syl 17 . 2 (𝐵 ∈ V → (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵})
4 reldmmap 8598 . . . 4 Rel dom ↑m
54ovprc1 7307 . . 3 𝐵 ∈ V → (𝐵m 𝐴) = ∅)
6 0ss 4335 . . 3 ∅ ⊆ {𝑓𝑓:𝐴𝐵}
75, 6eqsstrdi 3979 . 2 𝐵 ∈ V → (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵})
83, 7pm2.61i 182 1 (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1541  wcel 2109  {cab 2716  Vcvv 3430  wss 3891  c0 4261  wf 6426  (class class class)co 7268  m cmap 8589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-map 8591
This theorem is referenced by: (None)
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